Question

AB and CD are two parallel lines and a transversal l intersects AB at X and CD at Y. Prove that the bisectors of the interior angles form rectangle.

Answer 1

Given :-

AB║CD and transversal $l$ intersects AB at X and Cd at Y.

To be proof :-

PYQX is a rectangle

PROOF :

$\bf \angle{DYX}=\angle{AXY}$   [Alternate interior angles]

$\bf \frac{1}{3}\angle{DYX}=\frac{1}{2}\angle{AXY}$

∴ ∠1 = ∠2

Now,

XY intersects  PX and QY at X and Y respectively,

such that

∠1 = ∠2

∴ PX║QY

Similarly,

PY║QX

So,

PYQX is a parallelogram

Now,

∠BXY + ∠DYX = 180°   [consecutive interior angles]

Or

$\bf 2\angle{2}+2\angle{3}=180^o$

$\bf \angle{2}+\angle{3}=90^o$

⇒ ∠1 +∠3 = 90°    [ ∴ ∠2 = ∠1 ]

⇒ ∠QXP = 90°

∴ PYQX is a rectangle

Hence proved.

Answer 2

Step-by-step explanation:

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